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Q)

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

$\begin {array} {1 1} (A)\;\large\frac{x^2}{25}-\large\frac{y^2}{9}=1 & \quad (B)\;\large\frac{x^2}{9}+\large\frac{y^2}{25}=1 \\ (C)\;\large\frac{y^2}{25}-\large\frac{x^2}{9}=1 & \quad (D)\;\large\frac{x^2}{25}+\large\frac{y^2}{9}=1 \end {array}$ Comment
A)
Toolbox:
• Equation of an ellipse whose major axis is along the x - axis is given by $\large\frac{x^2}{a^2}$$+\large\frac{y^2}{b^2}$$=1$
• If a point move in a plane in such a way that the sum of its distances from two fixed points is constant, then the path is an ellipse . The constant value is equal to the length of the major axis of the ellipse.
Step 1 :
Let us take A and B as the two positions of the two flag posts and P be the position of the man.
$\therefore PA + PB = 10$
Hence it is clear that the path described by the man is an ellipse.
The points A and B are the foci.
The length of the major axis is 10 m.
Step 2 :
The equation of the ellipse is of the form $\large\frac{x^2}{a^2}$$+\large\frac{y^2}{b^2}$$=1$
Since $2a=10 \Rightarrow a = 5$
Distance between the foci is 2c = 8
$\therefore c = 4$
We know $c = \sqrt{a^2-b^2}$
or $c^2 = a^2-b^2$
Substituting the value for c and a we get,
$16 = 25-b^2$
$\therefore b^2=9 \Rightarrow b = 3$
Hence the equation of the path traced by the man is
$\large\frac{x^2}{25}$$+\large\frac{y^2}{9}$$=1$